SOLUTION: A pole leans away from the sun at an angle of 9° to the vertical. When the elevation of the sun is 49°, the pole casts a shadow 43 ft long on level ground. How long is the pole?

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Question 1102090: A pole leans away from the sun at an angle of 9° to the vertical. When the elevation of the sun is 49°, the pole casts a shadow 43 ft long on level ground. How long is the pole?
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Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Draw out the picture

Explation of points:
Point A is the base of the pole
Point B is the furthest point along the ground where the shadow extends (aka tip of the shadow)
Point C is the sun's location (keep in mind the diagram isn't to scale)
Point D is the tip of the pole

Based on those points we can say
AD is the pole itself
AB is the pole's shadow

So far, we don't know what x or y are. We'll need the value of y which is dependent on us finding x first.
To find x, we use the fact that angle BAC is a right angle (90 degree angle)
(angle DAC) + (angle BAD) = 90
9 + x = 90
9 + x-9 = 90-9
x = 81

Focus on triangle DBA. We just found x = 81 and we know that 49 is another angle of this triangle. The only unknown is y. Add up the three angles (x,y,49) and set this equal to 180. Then solve for y
angle1+angle2+angle3 = 180
x+y+49 = 180
81+y+49 = 180
y+(81+49) = 180
y+130 = 180
y+130-130 = 180-130
y = 50

So far, we found that x = 81 and y = 50 giving this updated picture

We want the length or height of the pole, so we want the length of segment AD

Use the law of sines to find the length of segment AD
Focus solely on triangle DBA (erase any unneeded segments and points if things get too cluttered/confusing)
Focusing on this triangle, we have
angle A = 81
angle D = 50
angle B = 49
side d = 43 (opposite angle D)
we want side b (opposite angle B)

Law of Sines
sin(D)/d = sin(B)/b
sin(50)/43 = sin(49)/b
b*sin(50) = 43*sin(49) cross multiply
b = 43*sin(49)/sin(50) divide both sides by sin(50)
b = 42.3637456561236 Use your calculator to compute the approximate value

The pole is roughly 42.3637456561236 feet long/tall.